Household Time Allocation

Simultaneous game, clarifies best response and Nash equilibrium.
Mixed Nash equilibria could also be considered.
Material is an Excel File
and a Gambit file.

Do you also have a roommate who doesn't seem to do his or her share of the work?
Their usual excuses are: "I am just not good with cleaning", or "I don't care about
how clean the room is". If you belive them, in both cases you are left spending
more hours cleaning your joint room than your
roommate, and we will see why this is rational and even unavoidable unless there is some
outside pressure.

Cleaning your room is not really a game, but we model it by a game.
It is important to understand that this modelling has to be done with care,
and you also should be cautious when drawing conclusions from your model.
This is also discussed in this chapter.

The first model

HOUSEHOLD TIME ALLOCATION(p): Ann and Beth(p) share a room. Both decide to spend 0, 1, 2, 3, or 4 hours
on Saturday mornings (between 8:00 and 13:00) on household matters. However, Beth(p) is a little clumsy
and cannot work as efficient as Beth. Every hour Beth(p)) works is equivalent to p hours work of Ann, where
p < 1. The (immaterial) payoff for each is the product of

the total time somebody worked on household matters (where Beth's time is multiplied by a factor
of p), and

the free time they have between 8:00 and 13:00

.

This is discrete version of an example given in [Baldani/Bradfield/Turner, 18.7].
A0, A1, ... A4 are the options for Ann to work for 0,1, ... 4 hours,
and B0, B1, ... B4 are the options for Beth to work for 0, 1, ... 4 hours.

Open this Excel sheet to see and change
the payoff matrices for different p. The best response values highlight automatically.

Playing around, you will observe that the best response digraph for p < 0.25 looks like
shown to the right. There are two Nash equilibria: (A2,B0) and (A3,B0).
Ann will work for 2 or 3 hours, and Beth will work not at all.

For 1/4 < p < 1/2 the best response digraph looks
. There are still the same
two pure Nash equilibria, (A2,B0) and (A3,B0).

For 1/2 < p < 2/3 the best response digraph looks
, producing the
two pure Nash equilibria (A2,B1) and (A3,B0). Beth may even work for an hour!

For 2/3 < p < 3/4 the best response digraph looks
,
but the two pure Nash equilibria (A2,B1) and (A3,B0) remain.

For 3/4 < p < 1 the best response digraph looks
.
The only pure Nash equilibrium now is (A2,B1).

If p increases, Beth works more and Ann works less.
The better Beth gets at doing household matters, the more time she will have to spend with it.

A more general model

0

1

2

3

4

5

0

A0,0,B0,0

A0,1,B0,1

A0,2,B0,2

A0,3,B0,3

A0,4,B0,4

A0,5,B0,5

1

A1,0,B1,0

A1,1,B1,1

A1,2,B1,2

A1,3,B1,3

A1,4,B1,4

A1,5,B1,5

2

A2,0,B2,0

A2,1,B2,1

A2,2,B2,2

A2,3,B2,3

A2,4,B2,4

A2,5,B2,5

3

A3,0,B3,0

A3,1,B3,1

A3,2,B3,2

A3,3,B3,3

A3,4,B3,4

A3,5,B3,5

4

A4,0,B4,0

A4,1,B4,1

A4,2,B4,2

A4,3,B4,3

A4,4,B4,4

A4,5,B4,5

5

A5,0,B5,0

A5,1,B5,1

A5,2,B5,2

A5,3,B5,3

A5,4,B5,4

A5,5,B5,5

Ann's utility is increasing in each row from left to right,
Ai,0 ≤ Ai,1 ≤ Ai,2 ≤ Ai,3 ≤ Ai,4 ≤ Ai,5,
and Beth's utility is increasing in each column from up to down as
B0,i ≤ B1,i ≤ B2,i ≤ B3,i ≤ B4,i ≤ B5,i.
Moreover, in each column, Ann's utility is increasing until it reaches the maximum of that row,
and decreases from there on.
The question is: What will happen? Do we necessarily have the same pattern as above?